No marginal arbitrage of the second kind for high production regimes in discrete time production-investment models with proportional transaction costs

We consider a class of production-investment models in discrete time with proportional transaction costs. For linear production functions, we study a natural extension of the no-arbitrage of the second kind condition introduced by M. Rasonyi [13]. We show that this condition implies the closedness of the set of attainable claims and is equivalent to the existence of a strictly consistent price system under which the evaluation of future production profits are strictly negative. This allows to discuss the closedness of the set of terminal wealth in models with non-linear production functions which may admit arbitrages of the second kind for low production regimes but not marginally for high production regimes.